PROP. XIII. The rectangle contained by the segments of a chord PQ which passes through a fixed point O bears a constant ratio to the square on the parallel semi-diameter CD.

Also the rectangles contained by the segments of any two intersecting chords are to one another as the squares of the parallel semi-diameters.

Let the semi-diameter CP bisect the chord in V and let qu be the ordinate of the point in which CO produced meets the curve.

Then, since QV, CV are the ordinate and abscissa of Q,

therefore

Also

CD2 – QV2 : CV2 = CD2 : CP2 [Prop. XII., Note,

= CD2 – qv2 : Cv, similarly.

OV2: CV2 = qv2 : Cv3,

[Euc. VI., 2,

since the ordinates QV, qu are parallel.

Therefore CD-QV+OV2: CV2=CD2 : Cv2. [Euc. v., 24.

But CV: Cv is equal to CO: Cq, which is a constant_ratio since 0, q are fixed points.

Hence CD-QV*+OV2: CD is a constant ratio.

Therefore QO.OR, being equal to QV2- OV2 (Euc. II., 5, Cor.), bears a constant ratio to CD2.

Again, take any other chord Q'R', passing through 0, and let CD' be the parallel semi-diameter.

Then, since 20.OR: CD is constant for all chords through 0, therefore

QO.OR: CD2 = Q'O.OR' : CD".

COR. 1. Let the chords move parallel to themselves until they become tangents. Then the rectangles become the squares of tangents drawn from an external point. Hence tangents drawn from the same point are as the parallel semi-diameters.

COR. 2. The ratio CD: CD' is constant for all pairs of chords parallel to QR and Q'R'. Hence the rectangles contained by the segments of any two intersecting chords are as the rectangles contained by the segments of any other two chords parallel to the former.

One or more of these chords may be supposed to become tangents as in Cor. 1.

COR. 3. In Cor. 2 let one pair of chords pass through the focus. Then, by Prop. XI., p. 13, the rectangles contained by the segments of any two intersecting chords are to one another as the lengths of the parallel focal chords.

PROP. XIV. If a circle and an ellipse intersect in four points their common chords will be equally inclined to the axis of the ellipse.

Let Q, R, Q, R' be the points of intersection and let QR cut Q'R' in 0.

Then the rectangles QO.OR, Q'O.OR' are as the squares on parallel semi-diameters of the ellipse. [Prop. XIII. But these rectangles are equal by a property of the circle. [Euc. III., 35. Hence the diameters parallel to QR, Q'R' are equal and therefore equally inclined to the axis.

It follows that QR, Q'R' are equally inclined to the axis. Similarly QR', Q'R and QQ', RR' are equally inclined to the axis.

PROP. XV. The tangent at P meets any diameter in T and

the conjugate diameter in t.

To prove that

PT.Pt=CD2,

where CD is the semi-diameter parallel to Pt.

Draw. PV, Dv, ordinates of the diameter CT, and PM an ordinate of Ct.

Let the tangent at D, which is parallel to CP, since CP, CD are conjugate, meet TC produced in t.

Then the rectangles CV.CT and Cv. Ct are equal to one another, since they are both equal to the square on the same semi-diameter. [Prop. IX.

But the ratio CV: Cv, that is PM: Cv, is equal to Pt: CD, by similar triangles PtM, CDv.

Therefore Pt: CD= Ct: CT, from above,

PROP. XVI. If a chord pass through a fixed point, the tangents at its extremities will intersect on a fixed straight line.

Draw CO, through the fixed point 0, to meet the curve in P, and let T be the point of intersection of tangents at the extremity of the chord which is bisected in 0.

T

Draw Cop, bisecting in o any chord through O, and meeting the curve in p.

Draw pU an ordinate of the diameter through O, and let Tt, drawn through T parallel to Up, (and therefore fixed), meet Cp in t. Let CO produced meet the tangent at Ρ in V. CO.CT=CP =CU.CV.

Then Therefore

CO: CV=CU: CT.

[Prop. IX.

But, by similar triangles, CO is to CV as Co to Cp, and CU to CT as Cp to Ct.

and the tangents at the extremities of the chord Oo intersect

in t.

Also Tt is a fixed straight line.

[Prop. IX. [Construction.

Note. As in the case of the parabola, O is called the pole

of Tt, and Tt the polar of 0.

PROP. XVII. If from any point t, tpp' be drawn to meet the ellipse in p, p', and the chord of contact of tangents through t in o, then tpop' will be cut harmonically.

Draw Cc to the middle point of pp' and produce it to meet the curve in Q.

Let the diameter PP' bisect in O the chord of contact of tangents through t.

Draw QV, an ordinate of this diameter, and let the tangent at meet the diameter in T.

Hence tc.totp.tp', or 2tp.tp'to (tp+tp'), since c is the middle point of pp'.

PROP. XVIII. The areas of the ellipse and auxiliary circle are as CB to CA.

Let P, Q be the adjacent points on the ellipse.

Produce the ordinates NP, MQ to meet the auxiliary circle in p, 4 respectively, and draw PQ, pq.